COMP SCI 2201 & 7201: Algorithm and Data Structure Analysis - IT Assignment Help

Assignment Task:

Task:

Exercise 1 Turing machines (15 points)
Develop a Turing machine for the language L = {w ∈ {0, 1} ∗| |w| is odd and w contains at least one 1}. This means that the Turing machine should accept all strings over {0, 1} of odd length that contain at least one 1. Show the state transition diagram in addition to your defi- nition of the Turing machine. Show the working of your Turing machine on the strings “0010”, “00101”, and “000”.

Exercise 2 (25 points) Minimum vertex cover The decision variant of the minimum vertex cover problem is stated as follows. Given an undirected graph G = (V, E) and an integer k. Is there a set V ′ ⊆ V of at most k nodes such that each edge is covered, i.e. e ∩ V′ ?= ∅, for all e ∈ E. Show that the decision variant of the minimum vertex cover problem is NP-complete. You may use that the decision variant of the maximum clique problem is NP-complete.
 

Exercise 3 (25 points) Maximum Independent Set The decision variant of the maximum independent set problem is stated as follows. Given an undirected graph G = (V, E) and an integer k. Is there a set V ′ ⊆ V of at least k nodes such that there is no edge between any pair of nodes, i.e. (V′ × V′ ) ∩ E = ∅ holds. Show that the decision variant of the maximum independent set problem is NP-complete by reducing the vertex cover problem to independent set.

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